SAGE: using CubeSats for Gravitational Wave Detection
Abstract
SAGE (SagnAc interferometer for Gravitational wavE) is a fast track project for a space observatory based on multiple 12U CubeSats in geostationary orbit. The objective of this project is to create a Sagnac interferometer with 73 000 km circular arms. The geometry of the interferometer makes it especially sensitive to circularly polarized gravitational waves at frequency close to 1Hz. The nature of the Sagnac measurement makes it almost insensitive to position error, allowing spacecrafts in ballistic trajectory. The light source and recombination units of the interferometer are based on compact fibered technologies, without the need of an optical bench. The main limitation would come from nongravitational acceleration of the spacecraft. However, conditionally upon our ability to postprocess the effect of solar wind, solar pressure and thermal expansion, we would detect gravitational waves with strains down to over a few days of observation.
SAGE: using CubeSats for Gravitational Wave Detection
Keywords: Space Mission, Nanosatellite, Gravitational waves, metrology
1 Introduction
Gravitationalwave (GW) astronomy will be a major observing window on the Universe in the next decades. The groundbased detectors that obtained the first direct detection of gravitational waves [abbott16] are sensitive around Hz, while the future Laser Interferometer Space Antenna (LISA) will reach maximum sensitivity at Hz. The gap in between these two ranges is and has been the subject of investigation by various groups, with quite a few spaceinterferometer proposals being published (for a review of gravitational wave detection in space, see reference [ni16]).
The traditional problem with space missions is the difficult science/cost ratio. Flagship missions like LISA can be funded thanks to the tremendous scientific return it will have on the community. Middle size/middle cost space missions can also exist, providing they give a consequent, but not risky, return on investment. However, low cost/small space missions have difficulties to exist since the science case is often limited, and the low cost implies increased risk. This is especially relevant in the context of GW detection. The space community will be focused for decades (and more) on the LISA mission, and it will be hard to find money for another mediumrange mission.
In this context, we still want to try to open the gravitational window for waves around 1 Hz. Attempts to go to lower frequencies with groundbased detectors are under study, but past the seismic noise, a ground detector will always be limited by Earth gravitational disturbances. Our solution to these difficulties is a low cost space mission. This is not a new concept: it has already been proposed to use offtheshelf satellites [tinto15]. However, with SAGE, we propose to go several steps further in simplification. The main simplifying step is the use of the CubeSat standard. But it is only possible because other steps are taken:

a low power laser (mW) at telecom wavelength (m)

fibered optics only, no bulk optics apart from the primary and secondary mirrors

free movement of the spacecraft (ballistic trajectory), no thrusters during science operation

because of 3., no free falling test mass, but post processing correction of external forces

volume limited to a 12U CubeSat (Figure 1)

because of 5., no sunradiation shielding
The main change, with respect to a simple scaling down of a LISAtype mission, is the absence of a Gravitation Reference Sensor (GRS). It means that the spacecraft must, by itself, act as a test mass. Post processing of the displacements of its center of mass will therefore be paramount.
2 Principle of the SAGE interferometer
The detector is based on 3 spacecrafts in geostationary orbit (GEO). Or more exactly slightly above the GEO orbit, in the socalled graveyard orbit [2005ESASP.587..373J]. Each spacecraft is at an altitude of around 36 000 km, with the full interferometer forming an equilateral triangle of side 73 000 km (Figure 2).
The interferometer will use time delayed heterodyne interferometry (TDI) [2014LRR....17....6T] at m. Each spacecraft is equipped with a single 200 mW fibered ECL laser. After dividing the beam thanks to a 50/50 fiber splitter, the two telescopes send two beams at 60 degrees from each other to the other two spacecrafts. For each spacecraft , there are two reference optical positions: and . These positions correspond to the two semireflective extremities of the single mode fibers (see Figure 3 for a representation of the optical layout inside the satellite). They are used as reference position because they are the last flat optical surfaces before collimation towards, respectively, spacecrafts and . They are purposely used to monitor the noncommon path in the optical setup between the two external metrology measurements (diode and in figure 3). This monitoring is made thanks to the diode .
On board each spacecraft, three optical path differences (OPD) are measured (denoted ). As mentionned in the previous paragraph, one OPD is measured on diode between the two reference positions inside the satellite (). Two other OPDs () are measured between the position of the outgoing fiber and the corresponding fiber on another satellite (with a delay equivalent to the time of flight). The TDI measurement is a combination of all the optical path measurements such that the absolute optical phase values become irrelevant. Here we consider an unequalarm configuration, a version of the Sagnac where the photons sweep through the interferometric arms 4 times. The below equations indicate the relations between OPD and optical phases at the reference positions, for satellite A. The optical length delay between two satellites are denoted for a photon going from spacecraft to . Note that because the interferometer is in a geostationary orbit, the equilateral triangle rotates at a speed of 360 deg each days. This orbital period results in a difference between and of almost a kilometer. Using the terms :
(1)  
(2)  
(3)  
(4)  
(5) 
The TDI measurement is a linear combination of the 5 measurements above. It is a Sagnac configuration in the sense that the measurement follows two beams in opposite directions:
(6) 
which, using Eqs.1 to (5), relates to the optical path lengths between the satellites by the following relation:
(7) 
In Eq (7), all the phase values have been canceled out. It means that the absolute phase of the laser beam does not matter, which is an important property of the Sagnac combination.
Another important advantage of the Sagnac configuration is that if the interferometer formation is constant (). A concrete example is the case of a satellite drifting on his orbit. The satellite average speed is of the order of . If the speed changes by 0.1% in a second, than the terms still stay below 3 m. This resolve one important problem of optical interferometry from space: the coherence length of the laser. Any measurement far away from the zero optical path would blur the fringes: wavelength instabilities translate into an optical path error. This phenomenon is also called the ”phase noise”. In the case of a geostationary constellation, it is especially relevant due to the speed of the satellites.
3 External forces
Because SAGE abandons free fall, no reference mass can be kept internally in a free falling state. We are thus loosing a precise reference position. Instead, the extremities of the fibers are the reference positions, and the whole satellite is the equivalent of the test mass. Any force from outside will affect the center of mass of the satellite, and hence, the OPD measurement.
The most important environmental forces in space are caused by solar radiation pressure and solar wind. Both noise terms have two components. A red component, which corresponds to variations in the particle density. The second component is a white noise caused by the shot noise of the particles.
But this is not all. The position of the barycenter of the satellite can also move due to thermal effects. This too has to be accounted for, especially in the context of expansion and deformation of the platform.
3.1 Solar Radiation Pressure
At 1 AU from the Sun the solar radiation averages to N/m. This acceleration is of the order higher than what is expected from the GW. However, most of the energy is at low frequency: the total solar irradiance (TSI) changes slowly with time. To characterize its power spectral density (PSD), we used data from the VIRGO (Variability of solar IRradiance and Gravity Oscillations) instrument onboard the SOHO observatory. VIRGO contains three channel sunphotometers (SPM) that monitor the irradiance at 402, 500, and 863 nm with a bandwidth of 5 nm.[Frohlich95, Frohlich97, 2002SoPh..209..247J].
In figure 4 we present a dataset that covers the full year 2013. Data sampling is 60 s, which means that the maximum frequency (Shannon’s frequency) is 8 mHz. We plotted the three PSDs for the 3 channels. One can clearly see the pressure modes that dominate the PSD around 3 mHz (resonance of the radiative structure). Below 1mHz, the spectrum is dominated by the gravitational modes (convective structure). The trend, below and after the modes, are caused by the granulation. This is a red noise which can be fitted by a logarithmic relation: . For respectively the blue, green, and red channels we have obtained: , and m and and . The difference in amplitude between the three channels is a consequence of the optical depth of the Sun as a function of the wavelength.
Weighting the PSDs by the solar 6000 K black body yields to the relation:
(8) 
At the same time, the shot noise can be deduced from the ratio between total energy (Wm) and the energy of a nm photon:
(9) 
of amplitude well below the irradiation variation.
3.2 Solar Wind
To estimate the amount of solar protons that will be accelerating the satellite, we restricted ourselves to the low speed ones () which are emitted by the Sun as the solar wind. This wind is monitored from the L1 Lagrange point by the SWEPAM instrument on board the ACE satellite. The instrument monitor both the speed and the density of the ions in the solar wind. The density is typically of the order a several particles per . However, it is especially sensitive to interplanetary shocks, which cause a sudden rise in density of particles (up to cm), followed by an acceleration of the wind. These spikes in the solar wind momentum dominate the power spectrum.
Figure 5 plots one year of data of ion density and speed. The force applied to the satellite, in N/m, is calculated from the mass of a proton : . The fit of the power density gives :
(10) 
while the shot noise can be derived from the average density ( per cm) and speed ( km/s) of protons :
(11) 
3.3 Temperature sensibility
The reference positions are the extremities of the fibers ( in figure 3). Even if the center of mass is stable, the position of the two reference positions may move with respect to it. The simplest example is a rotation of the satellite. This rotation is fortunately not an issue because it can be measured and postprocessed thanks to our knowledge of the optical orientation with respect to the two other spacecrafts.
However, a major problem will come for the thermal modification of the spacecraft, even if the full CubeSat is made of a material with a very low thermal expansion coefficient (eg, SiC). For example, with a material of thermal expansion of K, and considering the typical size of 10 cm, it would creates a noise of the order of 200 nm/K.
Engineering a design that makes the center of mass unaffected by temperature gradients will be a major challenge. In the worst case, it requires a monitoring of the spacecraft temperatures at a level of K, which means that the effect on the reference positions will be maintained below 20 pm.
4 Optomechanical design
4.1 The telescopes
Each satellite will manoeuvre to maintain the equilateral formation. Thus each satellite must be launching two beams at separation. For compactness, but most importantly, for robustness, the two telescopes are intertwined. The two primary mirrors, in Zerodur and of size 10 cm, are maintained together by molecular cohesion. The two secondary mirrors, also in Zerodur and of diameter 36 mm, are also similarly glued and maintained together by molecular cohesion. To maximize thermal stability, the two M2s are kept in position with respect to the M1s through a common structure in invar material.
The optomechanical concept of the instrument is presented in Figure 6. It is driven by the following constraints:

Stability of the angle of the two telescopes satellites to project an equilateral triangle in space

Thermal environment

Available space inside the 12U platform
Thermal environment and stable angular reference between the two telescopes of each satellite were the main reasons to choose a Zerodur + invar assembly instead of a single material for the structure and the optics. For thermal stability of the angular reference and space constraint, on axis optics was preferred at the cost of a central obstruction of 14% with a 100 mm pupil diameter. The two M1 and M2 assemblies are glued during the alignment phase to provide the reference angle. Molecular cohesion is then applied to the M1 assembly between the upper and lower zerodur interface plates. Both M1 and M2 units are then integrated and glued to the invar monobloc body of the telescope. The invariant point of the differential thermal expansion of the complete assembly is as close as possible to the M2 unit. An accurate temperature sensing all over the two telescopes assembly provides input to the model of the differential thermal expansion (Invar/Zerodur) on the reference pointing. This bias is taken into account in the correction and error budget of the full interferometer metrology.
The beams are launched from a single mode fiber at the focal point of each telescope. Fine pointing is achieved thanks to a piezo stage with range 400 m. This range, combined to the f/D value of 0.25, gives a field of view of . This means that during operation, the position of the satellites must be accurate to a projected distance of 73 km ().
4.2 Collimating and coupling the single mode light
The power of the source results from a difficult compromise. On one hand photons are important for a precise metrology measurement, so it is necessary to have a powerfull beam. But on the other hand the thermal stability, consumption level, and nonlinear effects in the fibers require moderate power.
To help with this compromise, it is important to maximize the throughput. This comes in two flavors: decreasing the collimating loss and increasing the coupling efficiency. Both are constrained by the mode of the fiber and the shape of the telescope aperture. The shadow from the spider arms and dualsecondary mirrors is presented on figure 7. This shadowing is important due to the fact that we wanted to keep the optics onaxis for thermal stability.
We can approximate the mode of the single mode fiber to be a 2 dimensional Gaussian. Then, the mode writes:
(12) 
where is the radial position in the pupil, and is the numerical aperture as defined by the flux at . The total throughput of the system is:
(13) 
where is the pupil geometry: 1 if the light passes through, 0 otherwise. The optimization of as a function of is shown in Figure 7. It has a maximum throughput flux for NA. The throughput is then 49.6%, meaning that 50.4% of the laser power will be lost and/or absorbed by the optical system.
Assuming the receiving satellite is sufficiently distant, we can expect the incoming field to be flat and constant over the pupil. Then, the coupling efficiency is:
(14) 
In the end, the important value is . It reaches a maximum of .
4.3 Diffraction analysis
The telescope should be large to minimize diffraction, and large to maximize the reception: the received flux increases as the power of 4 with the telescope size.
To calculate the far field flux at a distance of km, we can use the Fraunhofer approximation. The electromagnetic field is the Fourier transform of the field inside the emitting telescope pupil:
(15) 
where m. The resulting diffraction pattern is presented in the left panel of figure 8. Maximum is achieved on axis where the flux is around of the emitted flux per square meter. Integrated over the pupil, we get a throughput due to the diffraction of .
4.4 Photon error budget
The shot noise is basically the consequence of the quantum limit to measure the state of a photon. It depends on the total number of photons received by the satellite: the more photons we receive, the higher the accuracy on the phase measurement. Using a laser power of 200 mW (divided towards the two other satellites), the photon error depends on the total energy received at the other end. In our case:
(16) 
At wavelength m, the accuracy of the metrology measurement is limited by the photon noise:
(17) 
The strain of the most energetic gravitational waves is of the order . For an interferometer of arm length 73 000 km, the effect of the gravitational wave is 0.7 pm in amplitude. It seems small with respect to . However, the baseline concept is to fit inspiral patterns over several month, even years. In the example case of Fig 9, the signal can be detected with a signal to noise ratio of 8 in a day of science operation.
5 Conclusion
Can we really detect gravitational waves from space without a test mass? We believe so.
On one hand, it is true that a freefalling test mass allows to give a stringent reference point for a dragfree system. This may be the only solution for a high precision measurement. However, a dragfree system complicates the mission drastically: low noise thrust, internal metrology, complicated optical setup etc.
On the other hand, there is still precious science to be done regardless of the environmental effects. Maybe it is preferable to keep the satellite ”quiet”, and let it accelerate naturally. This is especially interesting for midfrequency observations, since the effect of external forces decreases as . However, it relies on the fact that environmental pressure forces can be measured and subtracted by postprocessing.
Acknowledgments
S. Lacour acknowledges support from ERC starting grant No. 639248. The VIRGO instrument onboard SoHO is a cooperative effort of scientists, engineers, and technicians, to whom we are indebted. SoHO is a project of international collaboration between ESA and NASA. Plot in Figure 9 was generated using the PyCBC software package [Canton:2014ena, Usman:2015kfa, Nitz:2017svb, alex_nitz_2018_1256897].